
for the Yang-Mills partition function on an oriented Riemann surface \(\Sigma _p\) 
of genus \(p\)  with defects labeled by
\(\varphi _1,\dots ,\varphi _p\in \mathsf {Out}(G)\)  around non-contractible
cycles of \(\Sigma _p\) . When all defects are trivial, \(\varphi _i=\text{id}_G\)  for
\(i=1,\dots ,p\) , this combinatorial expression is just the usual
Migdal-Rusakov heat kernel expansion for the partition function of Yang-Mills theory
on
\(\Sigma _p\)  [1]}, [2]}, [3]}, [4]}. In
general it agrees with the computation
of [5]} for the particular defect
configuration at hand.
